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1. The interior angles of a polygon are in AP. The smallest angle is 120° and the common differences is 5". Find the number of sides of the polygon.
- A. 7
- B. 7
- C. 7
- D. 7
Answer: Option C
Explanation:
To find the number of sides of the polygon, you can use the following formula for the sum of interior angles in a polygon:
Sum of Interior Angles = (n - 2) × 180°
Where "n" is the number of sides of the polygon.
In this case, you are given that the smallest angle is 120°, and the common difference between the interior angles is 5°. So, you can set up an arithmetic progression (AP) to find the sum of the interior angles:
First angle (a) = 120° Common difference (d) = 5°
The sum of an AP is given by:
Sum = n/2 * [2a + (n-1)d]
Now, plug in the values:
Sum = n/2 * [2(120°) + (n-1)(5°)]
Simplify:
Sum = n/2 * [240° + 5°(n-1)]
Sum = n/2 * [240° + 5n - 5°]
Sum = n/2 * [5n + 235°]
Now, you know that the sum of interior angles of the polygon is equal to (n - 2) × 180°:
n/2 * [5n + 235°] = (n - 2) × 180°
Now, solve for "n":
n[5n + 235°] = 2(n - 2) × 180°
5n^2 + 235n = 2(180n - 360)
5n^2 + 235n = 360n - 720
Rearrange:
5n^2 - 125n - 720 = 0
Now, you can solve this quadratic equation for "n." You can either use the quadratic formula or factor it. Factoring, you get:
(n - 9)(5n + 80) = 0
Setting each factor equal to zero:
- n - 9 = 0 => n = 9
- 5n + 80 = 0 => 5n = -80 => n = -16 (but this doesn't make sense in this context)
So, the number of sides of the polygon is 9.
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